The old 1858 Clarke model of earth used a radius of 6378293 meters, and the international 1972 model used a radius of 6378388 meters. I approximated the radius of the earth based upon the WGS84 model of the earths ellipsoid (which is the standard for GPS and navigational aids), where the semi major axis is 6378137 meters (I approximated with 6378.15 km).
To be more accurate one should factor in the inverse flattening (1/f) 1/298.257223563 for the shape of the ellipsoid, then things get complicated when you need to know the position of the station and the aircraft to determine the shape of the geodesic curve between them. If you are going to get that accurate them you need to look at masking by terrain, and the type of terrain the radio wave are propagating over.
So depending on what model of earth you use, you will get different numbers, 40nm @1000 ft, and 50 nm @ 1500 ft are about right.
1.23*sqrt(altitude) is close enough, and is easy to learn.
